Homework Homework Assignment #9 Review Section 5.8 Page 365, Exercises: 1 – 49(EOO) Quiz next time...

Homework

Homework Assignment #9 Review Section 5.8 Page 365, Exercises: 1 – 49(EOO) Quiz next time

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

Homework, Page 3661. A bacteria population P obeys the exponential growth law P(t) = 2,000e1.3t (t in hours).

(a) How many bacteria are present initially?

(b) At what time will there be 10,000 bacteria?


1.3 00 2000 2000*1 2,000 bacteriaP e

1.3 1.3 1.32000 10000 5 ln ln 5

ln 51.3 ln 5 1.238

1.3

There will be 10,000 bacteria at 1.238 hours.

t t tP t e e e

t t

t

Homework, Page 3665. The decay constant of Cobalt–60 is 0.13 years–1. What is its half-life?


ln 2 ln 20.13 5.332

0.13

half-life 5.332 years

k half lifek

Homework, Page 3669. Find the solution to y ′ = 3y satisfying y(2) = 4.


3 236

3

43 2 4

0.009915 0.009915

t

t

y y y Ce y Ce Ce

C y t e

Homework, Page 36613. Assuming that population growth is approximately exponential, which of the two sets of data is most likely to represent the population (in millions) of a city over a 5-year period?


Year 2000 2001 2002 2003 2004

Data I 3.14 3.36 3.60 3.85 4.11

Data II 3.14 3.24 3.54 4.04 4.74

Data II is most likely to represent the population of the city over the five-year period, as Data I is almost linear.

Homework, Page 36617. A 10-kg quantity of radioactive isotope decays to 3-kg after 17 years. Find the decay constant of the isotope.


17 1710 17 3 10 ln 3 ln10 ln

ln 3 ln10ln 3 ln10 17 ln 0.0708

17

Decay constant = 0.0708

k kktP t e P e e

k e k

Homework, Page 36621. The atmospheric pressure P(h) (in psi) at a height h (in miles) above sea level on earth satisfies a differential equation P′ = – kP for some positive constant k.

(a) Measurements with a barometer show that P(0) = 14.7 and P(10) = 2.13. What is the decay constant k?


0

10 10

0 14.7 14.7

14.7 10 2.13 14.7 ln 2.13 ln14.7 ln

ln 2.13 ln14.7ln 2.13 ln14.7 10 ln 0.19317

10

Decay constant = 0.19317

kkh

k kkh

P h Ce P Ce C

P h e P e e

k e k

Homework, Page 36621. (b) Determine the atmospheric pressure 15 miles above sea level.


0.19317 150.1931714.7 15 14.7 0.811

Atmospheric pressure at 15 miles = 0.811 psi

hP h e P e

Homework, Page 36625. In 1965, Gordon Moore predicted that the number of transistors on a microchip would increase exponentially.

(a) Does the table of data

confirm Moore’s prediction?

If so, estimate the growth constant.


Year No. Trans,

1971 2,250

1972 2,500

1974 5,000

1978 29,000

1982 120,000

1985 275,000

1989 1,180,000

1993 3,100,000

1997 7,500,000

1999 24,000,000

2000 42,000,000

The data seems to support Moore’s prediction.

Homework, Page 36625. (b) Plot the data in the table.


Year No. Trans,

1971 2,250

1972 2,500

1974 5,000

1978 29,000

1982 120,000

1985 275,000

1989 1,180,000

1993 3,100,000

1997 7,500,000

1999 24,000,000

2000 42,000,000

Homework, Page 36625. (c) Let N(t) be the number of transistors t years after 1971. Find an approximate formula N(t) ≈ Cekt, where t is the number of years after 1971.


7

7

18 18

29

0 2250 2250 7 29000 2250

ln 29000 ln 2250 ln ln 29000 ln 2250 7 ln

ln 29000 ln 22500.365

7

18 11800000 2250 ln1180000 ln 2250 ln

ln1180000 ln 22500.348

18

29 420000000 2250 ln 42000

kkt

k

k k

k

N N t e N e

e k e

k

N e e

k

N e

29000 ln 2250 ln

ln 42000000 ln 22500.339

29

ke

k

Homework, Page 36625. (d) Estimate the doubling time in Moore’s Law for the period 1971 – 2000.

(e) If Moore’s Law holds to the end of the decade, how many transistors will a microchip hold in 2010?


0.35 ln 22250 doubling time 1.98

0.35

doubling time 2 years

tN t e

0.35 3939 2250 1,906,787,060

39 1,910,000,000

N e

N

Homework, Page 36625. (e) Can Moore expect his prediction to hold indefinitely?

Moore cannot expect his prediction to hold indefinitely, as a some point transistors will get as small as they can, one or more molecules, in size. That coupled with the finite length of the connecting conductors will limit the miniaturization we have seen over the last several decades.


Homework, Page 36629. A certain quantity increases quadratically: P(t) =P0t2.

(a) Starting at time t0 = 1, how long will it take for P to double in size? How long will it take starting at t0 = 2 or 3?

(b) In general, starting at time t0, how long will it take for P to double in size?


2 2 20 0 0 0

2 2 20 0 0 0

2 2 20 0 0 0

1 1 2 2 2 1.414

2 2 4 8 8 2 2 2.828

3 3 9 18 18 3 2 4.243

P P P P t P P t t t

P P P P t P P t t t

P P P P t P P t t t

0 0In general, starting at , it will take until 2 for the size to double.t t

Homework, Page 36633. A bank pays interest at the rate of 5%. What is the yearly multiplier, if interest is compounded

(a) annually?

1.05

(b) three times per year?

(c) continuously?


3

30.051 1.016

3

0.05e

Homework, Page 36637. An investment increases in value at a continuously compounded rate of 9%. How large must the initial investment be to build up a value of $50,000 over a seven-year period?


0.09 7 0.63

0.63

50000 50000

5000026629.590 $26,659.59

rtA Pe Pe Pe

P Pe

Homework, Page 36641. If a company invests $2,000,000 to upgrade a factory, it will earn additional profits of $500,000 per year. Is the investment worthwhile, assuming an interest rate of 6%?


0.06 5

0.06 4 0.06 3 0.06 2

0.06 1

2000000 $2,699,717.61

500000 500000 500000

500000 500000

635624.57 598608.68 5637483.42 530918.27

+500000

$2,828,899.94

Since there will be m

rtA Pe A e

A e e e

e

ore money after five-years, the upgrades

are the better investment.

Homework, Page 36645. Use equation 3 to compute PV of an income stream paying out R(t) = $5,000/yr continuously for ten yr at r = 0.05.


0

1010 0.05 0.050 0

5000 5000 20

5000 12.131 20 $39,346.93

T rt

t t

PV R t e dt

PV e dt e

Homework, Page 36649. Show that PV of an investment that pays out R dollars/yr continuously for T years is R(1 – e–rt)/r where r is the interest rate.


00

0

Using Equation 3,

1 1 1

1

T

T rrt rt rT

rT

PV R t e dt R e R e er r r

eR

r

Homework

Homework Assignment #10 Review Sections 5.1 – 5.8 Page 369, Exercises: 1 – 97(EOO) Quiz next time


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